Étienne Ghys

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We define a notion of Ricci curvature in metric spaces equipped with a measure or a random walk. For this we use a local contraction coefficient of the random walk acting on the space of probability measures equipped with a transportation distance. This notions allows to generalize several classical theorems associated with positive Ricci curvature, such as(More)
In the plane R, we consider the sequence of points xi = (i, 0) (for i = 1, 2, . . .) and we denote by D(r) the disc of radius r, centered at the origin. In the space Xn of n-tuples of distinct points of D(n + 1/2), we consider the equivalence relation that identifies two n-tuples if one is obtained from the other by a permutation of the indices. We denote(More)
The trajectories of a vector field in 3-space can be very entangled; the flow can swirl, spiral, create vortices etc. Periodic orbits define knots whose topology can sometimes be very complicated. In this talk, I will survey some advances in the qualitative and quantitative description of this kind of phenomenon. The first part will be devoted to vorticity,(More)
A 2-dimensional predator-prey model with five parameters is investigated, adapted from the Volterra–Lotka system by a nonmonotonic response function. A description of the various domains of structural stability and their bifurcations is given. The bifurcation structure is reduced to four organising centres of codimension 3. Research is initiated on(More)
In this paper we study foliations F on compact manifolds M , of real codimension 2, with a transversal holomorphic structure. We construct a decomposition of M into dynamically defined components, similar to the Fatou/Julia sets for iteration of rational functions, or the region of discontinuity/limit set partition for Kleinian groups in PSL(2,C). All this(More)
Degenerate Riemannian metrics exist naturally in various contexts. Unfortunately, their study stops to the ‘admission of failure’ that they are too poor, for instance, to generate a coherent intrinsic or extrinsic differential geometry, e.g. a kind of Levi-Civita connection. In this first text, we start the investigation of rigidity aspects of these(More)
Theorem 1 in [2] gives a complete description of the situation on closed 3manifolds for which H(M ;O) = 0. On the other hand, Y. Carrière obtained in [3] a classification of riemannian foliations in dimension 3. Therefore, the association of theorem 1.1. and Brunella’s result gives a classification: the only transversely holomorphic foliations on closed(More)